This follows directly from the Schur complement formula (Section 0.8, [1]}). For any block matrix in which \(A\)  is invertible,
\( \begin{pmatrix} I & \mathbf {0} \\ -CA^{-1} & I \end{pmatrix} \begin{pmatrix} A & B \\ C& D \end{pmatrix}\begin{pmatrix} I & -A^{-1}B \\ \mathbf {0} & I \end{pmatrix} = \begin{pmatrix} A & \mathbf {0} \\ \mathbf {0} & D - CA^{-1}B \end{pmatrix}.\) 
